Although this is clearly on the wrong side of ethics/legality, an anonymous researcher built a botnet to scan the enire IPv4 address space. It's an interesting read.

Two years ago while spending some time with the Nmap Scripting Engine (NSE) someone mentioned that we should try the classic telnet login root:root on random IP addresses. This was meant as a joke, but was given a try. We started scanning and quickly realized that there should be several thousand unprotected devices on the Internet.

After completing the scan of roughly one hundred thousand IP addresses, we realized the number of insecure devices must be at least one hundred thousand. Starting with one device and assuming a scan speed of ten IP addresses per second, it should find the next open device within one hour. The scan rate would be doubled if we deployed a scanner to the newly found device. After doubling the scan rate in this way about 16.5 times, all unprotected devices would be found; this would take only 16.5 hours. Additionally, with one hundred thousand devices scanning at ten probes per second we would have a distributed port scanner to port scan the entire IPv4 Internet within one hour.